Independence of functions on a curve

Question

If you have a projective curve $C$ (feel free to add adjectives) over $\mathbb{F}_q$. Let $\{P_1, \ldots , P_n\}$ be the affine points and $\infty$ be the point at infinity (lets assume there is just the one point at infinity). If you have a linear independent set of function $\{f_1, \ldots,f_k\}$ in $L(t\infty)$. Is it true that the vectors $v_j = (f_j(P_1), \ldots, f_j(P_n))$ are linearly independent in $\mathbb{F}_q^n$?